Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

双参数广义Nielsen贝塔函数的不等式与渐近逼近

该文介绍了经典Nielsen贝塔函数的一种双参数推广,深入探讨了它的单调性、凹凸性、完全单调性与积分渐近性质,并在此基础上建立了一些新的不等式.

PROPER SPLITTINGS FOR RESTRICTED LINEAR EQUATIONS AND THE GENERALIZED INVERSE A_(T,S)~2

This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T （where, b ∈AT, and T is an arbitrary but fixed subspace of Cm） and the generalized in AT,S For the special case when b ∈AT and dim（T）=dim（AT）, this splitting iterative methverse AT,S hod converges to AT,Sb （the unique solution of the general restricted system Ax=bx ∈T）.

频率矫正广义S变换及在储层含油气性预测中的应用

随着我国油气勘探程度的不断加深,陆相页岩油气等非常规油气逐渐成为我国勘探开发的重点,陆相页岩非均质性强、储层薄且纵横向变化剧烈等典型特征导致含油气性预测精度差,影响后续井位部署和生产等.时频变换方法是储层含油气性预测最有效的工具.为了提高储层含油气性预测的精度,本文提出了一种频率矫正广义S变换(Frequency-Corrected Generalized S-Transform, FCGST),该变换既具有S变换多尺度多分辨率的特性,又可以通过自适应选取窗函数参数以控制窗函数的带宽和主频等,这一特性可以适用于不同工区地震数据的处理与解释.与传统的S变换和广义S变换不同,该变换修正了传统S变换和广义S变换中的尺度因子,改善了S变换和广义S变换等传统方法中时频谱主频向高频方向偏移的问题,有利于精确描述复杂储层含油气性.合成数据算例表明,频率矫正广义S变换提升了地震信号局部时频特征的表征能力;将频率矫正广义S变换应用于鄂尔多斯盆地某工区储层含油气性预测中,实钻结果数据进一步验证了该时频分析方法在储层含油气性解释中的有效性和潜力.

THE OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR SEIFFERT'S MEAN

For p ∈ R, the generalized logarithmic mean Lp（a, b） and Seiffert＇s mean T（a, b） of two positive real numbers a and b are defined in （1.1） and （1.2） below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp（a, b） 〈 T（a,b） 〈 Lq（a,b） holds for all a,b 〉 0 and a ≠ b.

GENERALIZED BOCHNER'S THEOREM FOR RADIAL FUNCTION

A radial function can be expressed by its generator through The positive definite of the function plays an important rote in the radial basis interpolation. We can naturally use Bochner's Theorem to check if is positive definite. This requires however a n-dhnensiotial Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-di-mensional space. The completely monotone function:, which is discussed in [4] is positive definite for arbitrary space dimensions. With this technique tve can very easily characterize the positive definite, of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, whcih are very useful in application. Can we find some conditions (as the completely monotone function) only for the \-dimen simial Fourier transform of the generator epto characterize a radial function 9, which is positivedefinite in n-dimensional (fixed n) spacel In this paper we defined a kind of incompletelymonotone function of order a, for a= 0,,1/2 ,1,3/2,(we denote the function class by ICM) ,in this sence a normal positvie function is in ICM a positive monotone decreasing function is inICM and a positive monotone decreasing and convex function is in ICM2- Based on this definition we get a generalized Bochner's Theorem for radial function-. If dimensional Fouriertransform of the generator of a radial function can be written as , then corre-spending radial function (x) is positive definite as a n-variate function iff F is an incomplete-ly monotone function of order a= (n- 1 )/2 (or simply In this way we have characterized the positive definite of the radial function as a n-vari-ate function through its generator in the sense of the Bocher's Theorem.

Generalized Extended tanh-function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher＇s equation, the nonlinear schr＆#168;odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.

Lie symmetries and conserved quantities for generalized Birkhoff system

To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the Lie symmetry of the system is derived. A con- served quantity of Hojman＇ s type and a Noether＇ s conserved quantity are deduced by the Lie symme- try under some conditions. One example is given to illustrate the application of the result.

LIMITING EXPRESSION FOR GENERALIZED INVERSE A_(T,S)^(2) AND ITS CORRESPONDING PROJECTORS

This paper presents the limiting expression for the gen calized inverse A T.S(2) and itscorresgonding projectors Since comonon imnortors inverses,such as and AT.S(2) etc are all generalized in e e AT.S(2) In fact,we give a unified limiting formula of computine such imporiant generalined inverses and its corresponding proiectors,Based on this we estalish and imbedling method fire compoting the generalized in verse AT.S(2) The results extend earlier work by various authors.

NOETHER’S CONSERVATION LAWS OF HOLONOMIC NONCONSERVATIVE DYNAMICAL SYSTEMS IN GENERALIZED MECHANICS

In the present paper, three kinds of forms for Noether’s conservation laws of hol-onomic nonconservative dynamical systems in generalized mechanics are given.

基于同步提取广义S变换的机械故障诊断方法研究

现有的同步提取变换(synchroextracting transform, SET)窗函数固定缺乏灵活性,在进行故障诊断时很难有效获取到高时频精度和高抗干扰性能的瞬时频率,针对此问题,结合广义S变换可以自适应调节窗函数宽度的优点,提出一种基于同步提取广义S变换(synchroextracting generalized Stransform, SEGST)的机械故障诊断方法。SEGST方法的特点在于将Rényi熵作为度量时频聚集性的标准,通过在高斯窗函数中引入2个尺度调节因子来选择参数的最佳值,对得到的广义S变换二维时频谱构造出同步提取算子来提取时频脊线处的时频系数,该算子能保留与信号的时变特征最相关的TF信息,剔除多余的模糊时频能量,从而得到高时频分辨率的时频能量特征。仿真结果表明,所提方法不论在时频分辨率方面,还是在噪声鲁棒性方面,都优于传统时频分析方法,并且保持了良好的重构性。最后,将所提方法应用于航空发动机高速滚动轴承故障诊断中,结果表明,该方法能够准确识别故障信号中的特征频率。

On the Application of the Adomian’s Decomposition Method to a Generalized Thermoelastic Infinite Medium with a Spherical Cavity in the Framework Three Different Models

A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity.A generalized set of governing equations is formulated in the context of three different models of thermoelasticity:the Biot model,also known as“coupled thermoelasticity”model;the Lord-Shulman model,also referred to as“generalized thermoelasticity with one-relaxation time”approach;and the Green-Lindsay model,also called“generalized thermoelasticity with two-relaxation times”approach.The Adomian’s decomposition method is used to solve the related mathematical problem.The bounding plane of the cavity is subjected to harmonic thermal loading with zero heat flux and strain.Numerical results for the temperature,radial stress,strain,and displacement are represented graphically.It is shown that the angular thermal load and the relaxation times have significant effects on all the studied fields.

ASYMPTOTIC EQUIVALENCE OF ALTERNATELY ADVANCED AND DELAYED DIFFERENTIAL SYSTEMS WITH PIECEWISE CONSTANT GENERALIZED ARGUMENTS

In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type （DEPCAG）. This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.

The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model

The nonlinear thermoelastic responses of an elastic medium exposed to laser generated shortpulse heating are investigated in this article. The thermal wave propagation of generalized thermoelastic medium under the impact of thermal loading with energy dissipation is the focus of this research. To model the thermal boundary condition(in the form of thermal conduction),generalized Cattaneo model(GCM) is employed. In the reference configuration, a nonlinear coupled Lord-Shulman-type generalized thermoelasticity formulation using finite strain theory(FST) is developed and the temperature dependency of the thermal conductivity is considered to derive the equations. In order to solve the time-dependent and nonlinear equations, Newmark’s numerical time integration technique and an updated finite element algorithm is applied and to ensure achieving accurate continuity of the results, the Hermitian elements are used instead of Lagrangian’s. The numerical responses for different factors such as input heat flux and nonlinear terms are expressed graphically and their impacts on the system’s reaction are discussed in detail.The results of the study are presented for Green–Lindsay model and the findings are compared with Lord-Shulman model especially with regards to heat wave propagation. It is shown that the nature of the laser’s thermal shock and its geometry are particularly determinative in the final stage of deformation. The research also concluded that employing FST leads to achieving more accuracy in terms of elastic deformations;however, the thermally nonlinear analysis does not change the results markedly. For this reason, the nonlinear theory of deformation is required in laser related reviews, while it is reasonable to ignore the temperature changes compared to the reference temperature in deriving governing equations.

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

基于同步提取增强广义S变换的柴油机气门性能退化状态评估

柴油机在运行过程中气门间隙逐渐增大,其状态会随气门性能退化而发生改变。针对传统状态评估方法难以对其气门性能退化状态进行准确评估的问题,提出基于同步提取增强广义S变换(Synchro Extracting Enhanced Generalized S-Transform,SEEGST)的柴油机气门性能退化状态评估方法。通过传感器采集反映柴油机状态的振动信号;为解决传统信号时频分析方法存在时频分辨率低、能量聚集性弱等问题,基于同步提取算法与广义S变换提出SEEGST时频分析方法,将振动信号转换为二维时频图;利用MLP-Mixer模型提取时频图像特征进行训练,实现柴油机状态评估。通过柴油机状态监测实验台开展气门性能退化实验,将所提方法与SSGST-MLPMixer、GST-MLPMixer、SEEGST-ViT、SEEGST-2DCNN、FFT spectrum-1DCNN 5种传统方法对比。实验结果表明:所提方法的整体评估准确率达到98.96%,可有效应用于柴油机气门性能退化状态评估领域,为开展柴油机气门性能退化状态评估提供一种新的思路。

基于广义S变换的光伏电站谐波和间谐波分析方法

针对S变换高斯窗函数形式固定不变的问题,引入高斯窗调节因子改进S变换,提出一种基于广义S变换的光伏电站谐波和间谐波分析方法。首先通过MATLAB/Simulink搭建光伏电站的仿真模型,在自然采样双极性SPWM调制方式下采集逆变器的三相输出电压;然后以A相电压为例,采用广义S变换对电压信号处理得到一个模时频矩阵;最后对该矩阵分析实现光伏电站谐波和间谐波参数的准确计算。仿真结果表明,本文方法的幅值计算误差最大仅为3.30×10^(-4)%,频率的平均计算误差为0%,远小于S变换幅值误差35.19%和频率误差2.39%;能满足光伏电站谐波和间谐波检测精度的需要。

Enhancing the resolution of seismic data based on the generalized S-transform

In this paper, we analyze the seismic signal in the time-frequency domain using the generalized S-transform combined with spectrum modeling. Without assuming that the reflection coefficients are random white noise as in the conventional resolution-enhanced techniques, the wavelet which changes with time and frequency was simulated and eliminated. After using the inverse S-transform for the processed instantaneous spectrum, the signal in the time domain was obtained again with a more balanced spectrum and broader frequency band. The quality of seismic data was improved without additional noise.

基于深度学习和广义S变换协同的风速预测

针对实测风速的非平稳性特点,提出一种基于深度学习和时频分析的风速混合预测方法。首先,采用经验模态分解(EMD)将风速分解为若干子层,由此得到趋势分量和脉动分量以降低风速的非线性。根据2个分量的时频特性,采用长短时记忆(LSTM)处理趋势分量,极限学习机(ELM)处理脉动分量。其次,引入广义S变换(GST)来获得预测过程中的时频特性。同时,采用改进的灰狼算法(IGWO)对GST、LSTM和ELM的参数进行优化。最后,以内蒙古某风场实测风速对所提模型进行验证,结果表明该模型具有较高的精度。